Efficient Numerical Solution Method for Large Deformation Analyses of Structures Based on the Updated Lagrangian Formulation
Publication: Journal of Aerospace Engineering
Volume 35, Issue 1
Abstract
Large deformation analyses of structures are of great importance to the evaluation of structural performance under extreme environmental loads, but currently available methods are time-consuming because of the requirement of factorizing large-scale matrices. The inelasticity-separated finite-element method (IS FEM), which can keep the global stiffness matrix unchanged and uses the Woodbury formula as the solver, was presented recently to provide a highly efficient tool for local material nonlinear analysis. To extend the high efficiency advantage of the IS FEM to large deformation analyses, in which the material nonlinearity may be nonlocal and the geometric nonlinearity should be considered, this paper proposes a novel numerical solution scheme by incorporating the updated Lagrangian (UL) formulation into the IS FEM framework. Within this scheme, a Woodbury approximation method (WAM) is introduced as an efficient solver, in which the changing global stiffness matrix is approximated as a constant matrix within a short time period, and a linear equation related to the Schur complement matrix is solved by the combined approximations (CA) method. To eliminate the additional error induced by the approximation, an adaptive iteration strategy (AIS) is presented, in which the approximation error involved in WAM solution is evaluated based on energy norm concept, and the global stiffness matrix is required to be updated adaptively according to the calculated error. The high efficiency and accuracy of the proposed method are finally demonstrated by the time complexity analysis and numerical examples.
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Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
The authors sincerely acknowledge the financial support from the National Key R&D Program of China (No. 2018YFC1504303) and the National Natural Science Foundation of China (No. 51878112). Any opinions, findings, and conclusions or recommendation expressed in this study are those of the authors and do not necessarily reflect the views of those acknowledged here.
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© 2021 American Society of Civil Engineers.
History
Received: Aug 29, 2020
Accepted: Aug 6, 2021
Published online: Sep 22, 2021
Published in print: Jan 1, 2022
Discussion open until: Feb 22, 2022
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